# Introduction

ecently there has been some interest in the notion of a locally closed subset of a topological space. According to Bourbaki [16] a subset S of a space (X, ) is called locally closed if it is the intersection of an open set and a closed set. Ganster and Reilly used locally closed sets in [13] and [14] to define the concept of LC-continuity, i.e. a function f : (X, ) (X, ) is LC-continuous if the inverse with respect to f of any open set in Y is closed in X. The study of semi open sets and semi continuity in topological spaces was initiated by Levine [6]. Bhattacharya and Lahiri [8] introduced the concept of semi generalized closed sets in topological spaces analogous to generalized closed sets which was introduced by Levine [5]. Throughout this paper, the word "space " will mean topological space The collections of semi-open, semi-closed sets and ?-sets in ( X, ) will be denoted by SO (X, ) , SC (X, ) and respectively. Njastad [7] has shown that is a topology on X with the following properties: and if and only if N where and N is nowhere dense in (X, ). Hence if and only if every nowhere dense (nwd) set in (X, ) is closed, therefore every transitive map implies ?-transitive. Also if every ?-open set is locally closed then every transitive map implies ?transitive; and this structure also occurs if ( X, ) is locally compact and Hausdorff [36, p. 140, Ex. B]] and every ?-open set is locally compact, then every ?-open set is locally closed.

In 1943, Fomin [27] introduced the notion of ?continuous maps. The notions of ?-open sets, ?-closed sets and ?-closure where introduced by Veli?cko [19] for the purpose of studying the important class of H-closed spaces in terms of arbitrary fiber-bases. Dickman and Porter [20], [21], Joseph [22] and Long and Herrington [31] continued the work of Velic ?ko. We introduce the notions of ?-type transitive maps, ?-minimal maps and show that some of their properties are analogous to those for topologically transitive maps. Also, we give some additional properties of ?-irresolute maps. We denote the interior and the closure of a subset A of X by Int(A) and Cl(A), respectively. By a space X, we mean a topological space (X, ) A point x ? X is called a ?-adherent point of A [19], if for every open set V containing x. The set of all ?-adherent points of a subset A of X is called the ?-closure of A and is denoted by . A subset A of X is called -closed if . Dontchev and Maki [22] have shown that if A and B are subsets of a space (X, ), then also Note also that the ?-closure of a given set need not be a -closed set. But it is always closed. Dickman and Porter [20] proved that a compact subspace of a Hausdorff space is ?-closed. Moreover, they showed that a ?-closed subspace of a Hausdorff space is closed. Jankovi´ [25] proved that a space (X, ) is Hausdorff if and only if every compact set is ?-closed. The complement of a ?-closed set is called a ?-open set. The family of all ?-open sets forms a topology on X and is denoted by or topology.

This topology is coarser than and that a space (X, ) is regular if and only if [26]. Then we observe that every theta-type transitive maps is transitive if (X, ) is regular. In general, will not be the closure of A with respect to (X , ). It is easily seen that one always has 
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , ? ? ? ? ? ? ) ( ? ? ? S U S \ ? ? ? U ) )) ( ( . . ( ? ? N Cl Int e i ? ? ? ? ? ? ? ? ? ? ) (V Cl A ) (A Cl ? ) (A Cl A ? ? ? ? (B) Cl (A) Cl = B) (A Cl ? ? ? ? ? (B) Cl (A) Cl = B) (A Cl ? ? ? ? ? . ? ? ? ? ? ? ? ? ? ? ? ? ) (A Cl ? ? ? ? ? ? A A Cl A Cl A Cl A ? ? ? ? ) ( ) ( ) (
where denotes the closure of A with respect to (X, ).

It is also obvious that a set A is ?-closed in (X, ) if and only if it is closed in (X, ). The space (X, ) is called sometimes the semi regularization of (X, ). A function is closure continuous [29] (? continuous) at x ? X if given any open set V in Y containing f(x), there exists an open set U in X containing x such that . [29] In this paper, we will study the relationship between new classes of topological transitive maps called type transitive and -type transitive, also, new classes of -type chaotic maps and -type chaotic maps. We have shown that every ?-type transitive map is a ?-type transitive map, but the converse not necessarily true and that every ?-type chaotic map is ?-type chaotic map, but the converse not necessarily true we will also study some of their properties.


# II.


# Preliminaries and Definitions

In this section, we recall some of the basic definitions. Let X be a space and A X . The intersection (resp. closure) of A is denoted by Int(A) (resp. Cl(A).

? Definition 2.1 [6] A subset A of a topological space X will be termed semi-open (written S.O.) if and only if there exists an open set U such that .

? Definition 2.2 [8] Let A be a subset of a space X then semi closure of A defined as the intersection of all semi-closed sets containing A is denoted by sClA.

? Definition 2.3 [9] Let (X, ) be a topological space and ? an operator from to ?(X) i.e ?: ? ? ?(X), where ?(X) is a power set of X. We say that ? is an operator associated with if for all

? Definition 2.4 [10] Let (X, ) be a topological space and ? an operator associated with ?. A subset A of X is said to be ?-open if for each x Ñ?" X there exists an open set U containing x such that . Let us denote the collection of all ?-open, semi-open sets in the topological space ( ) by , SO( ), respectively. We then have
SO . A subset B of X is said to be ?-closed [7] if its complement is ?-open.
? Definition 2.5 [9] Let (X, ) be a space. An operator ? is said to be regular if, for every open neighborhoods U and V of each x Ñ?" X, there exists a neighborhood W of x such that Note that the family of ? -open sets in (X, ) always forms a topology on X, when ? is considered to be regular finer than .

? Theorem 2.6 [30] For subsets A, B of a space X, the following statements hold: (1) , where D(A) is the derived set of A (2) If , then

Note that the family of ? -open sets in (X, ) always forms a topology on X denoted ?-topology and that ?-topology coarser than .

? Definition 2.7 [4]: Let A be a subset of a space X. A point x is said to be an -limit point of A if for each -open U containing x, . The set of all -limit points of A is called the -derived set of A and is denoted by [4] For subsets A and B of a space X, the following statements hold true: 1) where is the derived set of A 2) if then 3) 4)
? Definition 2.8
? Definition 2.9 [10]: The point x Ñ?" X is in the -closure of a set A X if ?(U) ? A??, for each open set U containing x. The -closure of a set A is the intersection of all -closed sets containing A and is denoted by Cl (A) . ? Remark 2.10: For any subset A of the space X,

? Definition 2.11 [10] Let (X, ) be a topological space. We say that a subset A of X is -compact if for every -open covering ? of A there exists a finite subcollection of ? such that I Properties of ? -compact spaces have been investigated by Rosa, E etc. and Kasahara, S [9,10].The following results were given by Rosas, E etc. [9].


# ?

Let (X, ) be a topological space and ? an operator associated with ?. A X and K A. If A is ?-compact and K is ? -closed then K is ? -compact.

? Theorem2.13 Let (X, ) be a topological space and ? be a regular operator on ?. If X is ? -2 T (see Rosa, E etc. and Kasahara, S) [9,10] and K X is ? -compact then K is ? -closed.

? Definition 2.14 [10] The intersection of all ? -closed sets containing A is called ? -closure of A, denoted by

? Remark 2.15 For any subset A of the space X, A ? Lemma2. 16 For subsets A and of a space (X, ) , the following hold:
1) A 2) closed; 3) If A B then 4)

# 5)

? Lemma 2.17 The collection of ? -compact subsets of X is closed under finite unions. If ? is a regular operator and X is an ?-2 T space then it is closed under arbitrary intersection.

Global Journal of Researches in Engineering ( )
F Volume XIV Issue V Version I 26 Year 2014 © 2014 Global Journals Inc. (US) ? A ? ? ? ? ? ? ? ? f : Y X? ) ( )) ( ( V Cl U Cl f ? ? ? ? ? ? ) (U Cl A U ? ? ? ? ? U ?a (U) U Ñ?" ? ? ? a(U) A ,? X ? ? ? ) (? ? ? ? ? ? ? ? ?(U) a(W) a(V).. ? ? ? ? ) ( ) ( A D A D ? ? B A ? ) ( ) ( B D A D ? ? ? . ) ( ) ( ) ( B A D B D A D ? ? ? ? ? ? ) ( ) ( ) ( B D A D B A D ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \ ( x A U ? ? ) (A D ? . ) ( ) ( A D A D ? ? D(A) B A ? ) ( ) ( B D A D ? ? ? ) ( ) ( ) ( B A D B D A D ? ? ? ? ? ? ) ( )) ( ( A D A A D A D ? ? ? ? ? ? ? ? ? ? ? ) ( ) ( A Cl A Cl A ? ? ? ? ? ? } ..., , 2 , 1 { n C C C i n i C A 1 ? ? ? . ? ? ? ? ? ) (A Cl ? . ? Cl(A) ? ) (A Cl ? . i A (i Ñ?" I) ? ? ) (A Cl ? ) (A Cl ? ) ( )) ( ( A Cl A Cl Cl ? ? ? ? ? ) (A Cl ? ? ) (B Cl ? ) : ) ( ( )) : ( ( I i A Cl I i A Cl i ? ? ? ? ? ? ? ) : ) ( ( )) : ( ( I i A Cl I i A Cl i ? ? ? ? ? ? ?
Relationship between New Types of Transitive and Chaotic Maps Theorem2.12 ? Theorem 2.21 [4]: For any subset A of a space X,

? Theorem 2.22. [4]: For subsets A, B of a space X, the following statements are true:
1) int (A ) is the largest ? -open contained in A 2) 3) If 4) 5)
? Lemma 2.23 [7] For any ? -open set A and any ?-closed set C, we have 1)


# 2)

3) ? Theorem 2.26 Let (X, f )and (Y, g) be two topological systems, if and are topologically ?r-conjugate. Then (1) f is topologically ?-transitive map if and only if g is topologically ?-transitive map;
? Remark 2.
(2) f is ?-type chaotic map if and only if g is ?-type chaotic map;

(3) f is ?-type chaotic map if and only if g is ?-type chaotic map.


# III.


# Transitive and Minimal Systems

Topological transitivity is a global characteristic of dynamical systems. By a dynamical system ( , ) [15] we mean a topological space X together with a continuous map . The space X is sometimes called the phase space of the system. A set is called inveriant if . A topological system (X, f ) is called minimal if X does not contain any nonempty, proper, closed inveriant subset. In such a case we also say that the map itself is minimal. Thus, one cannot simplify the study of the dynamics of a minimal system by finding its nontrivial closed subsystems and studying first the dynamics restricted to them.

Given a point x in denotes its orbit (by an orbit we mean a forward orbit even if is a homeomorphism) and denotes its ? -limit set, i.e. the set of limit points of the sequence .

The following conditions are equivalent:
? is ?-minimal (resp. ?-minimal),
? every orbit is ?-dense (resp. ?-dense) in X ,

? for every A minimal map is necessarily surjective if X is assumed to be Hausdorff and compact.

Now, we will study the Existence of minimal sets. Given a dynamical system , a set is called a minimal set if it is non-empty, closed and invariant and if no proper subset of A has these three properties. So, is a mi nimal set if and only if is a minimal system. A system is minimal if and only if X is a minimal set in . The basic fact discovered by G. D. Birkhoff is that in any compact system there are minimal sets. This follows immediately from the Zorn's lemma.

Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. This is how compact minimal sets may appear in non-compact spaces. Two minimal sets in either are disjoint or coincide. A minimal set A is strongly inveriant , i.e.

Provided it is compact Hausdorff. Let be a topological system, and ?r-homeomorphism of X onto itself. For A and B subsets of X, we let

We write for a singleton thus For a point we write for the orbit of x and for the ?-closure of . We say that the topological system is ?type point transitive if there is a point with ?dense. Such a point is called ?-type transitive. We say that the topological systems is topologically ?-type transitive (or just ?-type transitive) if the set is nonempty for every pair U and V of nonempty ?-open subsets of X.

? Theorem 2.8 [37] Let be a topological system where X is a non-empty locally ?-compact Hausdorff topological space and is ?-irresolute map and that X is ?-type separable. Suppose that f is topologically ?-type transitive. Then there is an element such that the orbit is ?-dense in X.


# a) Topologically ?-Transitive Maps

In [35], we introduced and defined a new class of transitive maps that are called topologically ?transitive maps on a topological space (X, ), and we studied some of their properties and proved some Global Journal of Researches in Engineering ( )
F Volume XIV Issue V Version I 27 Year 2014 © 2014 Global Journals Inc. (US) ? U U A : { ) ( ? ? ? A U ? }. A A ? ) ( ? . ? ? Y X f ? : ) ( 1 H f ? ) ( ) ( A Cl A A Cl ? ? ? ? ? ) ( int )) ( (int int A A ? ? ? ? ? B then ) ( int ) ( int B A ? ? ? A ) ( int ) ( int ) ( int B A B A ? ? ? ? ? ? ) ( int ) ( int ) ( int B A B A ? ? ? ? ? ? ) ( ) ( A Cl A Cl ? ? ) int( ) ( int C C ? ? int A Cl ? ? ? )) ( int( A Cl ? ={ d}, X}.
? ,{c, ? ,{c, d},{b, c, d},
? {a, c, d},X} X X f ? : Y Y g ? : f X X X f ? : X A ? f ? A A f ? ) ( f ? f X, ...} ), ( ), ( , { ) ( 2 x f x f x x O f ? ) (x f ? f ... ), ( ), ( , 2 x f x f x ) , ( f X X x f ? ) ( ? x ? X. . f ) , ( f X X A ? X A ? ) , ( A f A ) , ( f X ) , ( f X ) , ( f X ) , ( f X f ? A A f ? ) ( . ) , ( f X X X f ? : } ) ( : { ) , ( ? ? ? ? ? B A f n B A N n Z ) , ( ) , ( B x N B A N ? } {x A ? } ) ( : { ) , ( B x f n B x N n ? ? ? Z X x? } : ) ( { ) ( Z ? ? n x f x O n f )) ( ( x O Cl f ? ) (x O f ) , ( f X X x ? ) (x O f ) , ( f X ) , ( V U N ) , ( f X X X f ? : X x ? } : ) ( { ) ( N ? ? n x f x O n f
? results associated with these new definitions. We also defined and introduced a new class of ?-minimal maps. In this paper we discuss the relationship between topologically ?-transitive maps and ?-transitive maps. On the other hand, we discuss the relationship between ?-minimal and ?-minimal in topological systems.

? Definition 3.1.1 Let (X, ) be a topological space. A subset A of X is called ?-dense in X if .

? Remark 3.1.2 Any ?-dense subset in X intersects any ?-open set in X.

Proof: Let A be an ?-dense subset in X, then by definition, , and let U be a non-empty ?-open set in X. Suppose that A?U=? . Therefore is ?-closed and i.e.

B, but , so B, this contradicts that U
? Definition 3.1.3 [12] A map is called ?- irresolute if for every ?-open set H of Y, is ?-open in X.
? Example 3.1.4 [35] Let (X, ?) be a topological space such that X={a, b, c, d} and ? ={?, X ,{a, b}, {b}}. We have the set of all ?-open sets is ?(X, ?)={?, X, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}}and the set of all ?-closed sets is ?C(X, ?)={?, X, {c, d, {a, c, d}, {a, d}, {}a, c}, {d}, {c}}. Then define the map f : X?X as follows f ? Definition 3.1.5 A subset A of a topological space (X, ) is said to be nowhere ?-dense, if its ?-closure has an empty ?-interior, that is, .
(a)= a, f(b)= b, f(c)= d, f(d)= c, we have f is ?-irresolute because {b} is ? open and f-1({b})={b} is ?-open; {a,
? Definition 3.1.6 [35] Let (X, ) be a topological space, be ?-irresolute map then f is said to be topological ?-transitive if every pair of non-empty ?-open sets U and V in X there is a positive integer n such that . In the forgoing example 3.1.4: we have f is ?-transitive because b belongs to any non-empty ?-open set V and also belongs to f(U) for any ?-open set it means that so f is . ? -transitive.

? Example 3.1.7 Let (X, ?) be a topological space such that X ={a, b, c}and ?={?, {a}, X}.Then the set of all ?-open sets is ??={?, {a}, {a, b}, {a, c}, X}. Define f : X?X as follows f(a)=b, f(b)=b, f(c)=c. Clearly f is continuous because {a} is open and f({a})=? is open. Note that f is transitive because f({a})={b} implies that f({a})?{b}??. But f is not ?-transitive because for each n in N , fn({a})?{a, c}=?; since fn({a})={b} for every n ? N, and {b}?{a, c}=?. So we have f is not ?-transitive, so we show that transitivity not implies ?-transitivity.

? Definition 3.1.8 Let (X, ?) be a topological space. A subset A of X is called ?-dense in X if ? Remark 3.1.9 [38] Any ?-dense subset in X intersects any ?-open set in X.

Proof: Let A be a ?-dense subset in X, then by definition, ? Definition 3.1.11 A subset A of a topological space (X, ) is said to be nowhere ?-dense, if its ?-closure has an empty ?-interior, that is,

? Definition 3.1.12 [34] Let (X, ) be a topological space, and .-irresolute) map, then is said to be topologically ?-type transitive map if for every pair of ?-open sets U and V in X there is a positive integer n such that

We introduced a new definition on ?minimal [35] (resp. ?-minimal [34]) maps and we studied some new theorems associated with these definitions. Given a topological space X, we ask whether there exists ?-irresolute (resp. ?-irresolute) map on X such that the set , called the orbit of x and Global Journal of Researches in Engineering ( ) F Volume XIV Issue V Version I
![Journals Inc. (US)](image-2.png "")



			Year 2014 © 2014 Global Journals Inc. (US)
			© 2014 Global Journals Inc. (US) denoted by O ( x ) f , i s ?dense(resp. ?dense) in X for each x Ñ?" X.. A partial answer will be given in this section. Let us begin with a new definition.
			© 2014 Global Journals Inc. (US)
		
		
			

Relationship between New Types of Transitive and Chaotic Maps IV.


## Alpha-Minimal Functions

Associated with this new definition we can prove the following new theorem.

? Theorem 3.1.13 [35]: Let (X, ) be a topological space and be ? -irresolute map. Then the following statements are equivalent: (1) f is topological ?-transitive map ? Theorem 3.1.14 : [34] Let (X, ) be a topological space and be ? -irresolute map. Then the following statements are equivalent:

in X.

? Definition 4.1 (?-minimal) Let X be a topological space and f be ?-irresolute map on X with ?-regular operator associated with the topology on X. Then the dynamical system (X, f) is called ?-minimal system (or f is called ?-minimal map on X) if one of the three equivalent conditions [35] hold:

1) The orbit of each point of X is ?-dense in X.

2) for each x Ñ?" X 3) Given x Ñ?" X and a nonempty ?-open U in X, there exists nÑ?" N such that ? Theorem 4.2 [35] For (X, ) the following statements are equivalent:


## Topological Systems and Conjugacy

In this section, I introduce and define ?r-conjugated topological systems (X, f ) and (Y, g), where X and Y are almost regular topological spaces. First I will define ?r-homeomorphism and then I will prove new theorem associated with these new definitions:

? Definition 5.1 A map.is said to be -homeomorphism if is bijective and thus invertible and both and are ?rirresolute

? Definition 5.2 Two topological systems (X, f ) and (Y, g) are said to be almost regular systems if X and Y are almost regular topological spaces.

? Definition [38] 5.3 Let (X, f ) and (Y, g) be two almost regular systems, then and are said to be topologically ?r-conjugate if there is ?rhomeomorphism such that . will call h a topological ?r-conjugacy. Thus, the two almost regular topological systems with their respective function acting on them share the same dynamics VI.


## New Types of Chaos of Topological Spaces

We will give a new definition of chaos for ?-irresolute (resp. ?-irresolute) self map of a locally compact Hausdorff topological space X, so called ?-type chaos (resp. ?-type chaos). These new definitions imply John Tylar definition which coincides with Devanney's definition for chaos when the topological space happens to be a metric space, but not conversely.

? Definition 4.1 Let (X , f ) be a topological system, the dynamics is obtained by iterating the map. Then, f is said to be ?-type chaotic (resp. ?-type chaotic) on X provided that for any nonempty ?-open (resp. ?open) sets U and V in X, there is a periodic point such that and .

? Proposition 4.2 Let (X, f ) be a topological system.

The map f is ?-type chaotic (resp.?-type chaotic) on X if and only if f is ?-type transitive (resp. ?-type transitive) and the set of periodic points of the map f is dense (resp. dense) in X.

Let us prove only for ?-type chaotic Proof: I f f is ?-type chaotic on X, then for every pair of nonempty ?-open sets U and V, there is a periodic orbit intersects them; in particular, the periodic points are ?-dense in X. Then there is a periodic point p and with x ? U and y ? V and some positive integer n such that , so that therefore that is, f is ?-type transitive map.

The ?-type transitivity of f on X implies that for any nonempty ?-open subsets U, V ? X, there is n such that for some x ? U, Now define

. Then W is ?-open and nonempty with the property that . But since the periodic points of f are ?-dense in X, there is a p ? W such that . Therefore, and , so that f is ?-type chaotic map.


## VII.


## Conclusion

We have the following results

? Proposition 7.1. Every topologically ?-type transitive map is a topologically ?-type transitive map which implies topologically transitive map, but the converse not necessarily true.. ? proposition 7.2.Every ?-minimal map is ?-minimal map which implies minimal map, but the converse not necessarily true.. ? Theorem 7.3 Let (X, f )and (Y, g) be two topological systems, if and are topologically ?rconjugate. Then (1) f is topologically ?-transitive map if and only if g is topologically ?-transitive map;

(2) f is ?-type chaotic map if and only if g is ?-type chaotic map;

(3) f is ?-type chaotic map if and only if g is ?-type chaotic map.

? Proposition 7.4 Let (X, f ) be a topological system.

The map f is ?-type chaotic (resp.?-type chaotic) on X if and only if f is ?-type transitive (resp. ?-type transitive) and the set of periodic points of the map f is dense (resp.

dense) in X.

Global Journal of Researches in Engineering ( )
			
			

				


* 
	
		
		
	
	
		of Electrical and Electronic Science
		
			1
			1
			
			2014. 2014
		
	
	Published online March




* 
	
		
			GFArenas
		
		
			JDontchev
		
		
			LMPuertas
		
		Some covering properties of the ?-topology
				
			1998
		
	




* 
	
		On space with hereditarily compact ?-topologies
		
			MCaldas
		
		
			JDontchev
		
	
	
		Acta. Math
		
	




* 
	
		A note on some applications of ?-open sets
		
			MCaldas
		
	
	
		UMMS
		
			2
			
			2003
		
	




* 
	
		Generalized closed sets in topology
		
			NLevine
		
	
	
		Rend. Cire. Math. Paler no
		
			2
			
			1970
		
	




* 
	
		Semi open sets and semi continuity in topological spaces
		
			NLevine
		
	
	
		Amer. Math. Monthly
		
			70
			
			1963
		
	




* 
	
		On some classes of nearly open sets
		
			NOgata
		
	
	
		Pacific J. Math
		
			15
			
			1965
		
	




* 
	
		Semi-generalized closed sets in topology
		
			PBhattacharya
		
		
			KBLahiri
		
	
	
		Indian J. Math
		
			29
			
			1987
		
	




* 
	
		Operator-compact and Operatorconnected spaces
		
			ERosas
		
		
			JVielina
		
	
	
		Scientific Mathematica
		
			1
			2
			
			1998
		
	




* 
	
		Operation-compact spaces
		
			SKasahara
		
		
	




* 
	
		
	
	
		Mathematica Japonica
		
			24
			
			1979
		
	




* 
	
		Semi -topological properties
		
			GSCrossley
		
		
			SKHildebrand
		
	
	
		Fund. Math
		
			74
			
			1972
		
	




* 
	
		On ?-irresolute mappings
		
			NSMaheshwari
		
		
			SSThakur
		
	
	
		Tamkang J. Math
		
			11
			
			1980
		
	




* 
	
		A decomposition of continuity
		
			MGanster
		
		
			ILReilly
		
	
	
		Acta Math. Hungarica
		
			56
			3-4
			
			1990
		
	




* 
	
		Locally closed sets and LCcontinuous functions
		
			MGanster
		
		
			ILReilly
		
	
	
		Internat. J. Math. Math. Sci
		
			12
			3
			
			1989
		
	




* 
	
		
		
			NBourbaki
		
	
	
		General Topology Part
		
			1
			1966
		
	




* 
	
		H-closed topological spaces. (Russian) Mat. Sb
		
			NVVeli?cko
		
	
	
		English transl. Amer. Math. Soc. Transl
		
			70
			112
			
			1966. 1968
		
	




* 
	
		?-closed subsets of Hausdorff spaces
		
			RFDickman
		
		
			Jr
		
		
			JRPorter
		
	
	
		Pacific J. Math
		
			59
			
			1975
		
	




* 
	
		Porter, ?-perfect and ?-absolutely closed functions
		
			RFDickmanJr
		
		
			JR
		
	
	
		Ilinois J. Math
		
			21
			
			1977
		
	




* 
	
		Groups of ?-generalized homeomorphisms and the digital line
		
			JDontchev
		
		
			HMaki
		
	
	
		Topology and its Applications
		
			20
			
			1998
		
	




* 
	
		Weak and strong forms of ?-irresolute functions
		
			MGanster
		
		
			TNoiri
		
		
			ILReilly
		
	
	
		J. Inst. Math
		
	




* 
	
		
	
	
		Comput. Sci
		
			1
			1
			
			1988
		
	




* 
	
		On some separation axioms and ?-closure
		
			DSJankovic
		
	
	
		Mat. Vesnik
		
			32
			4
			
			1980
		
	




* 
	
		?-regular spaces
		
			DSJankovic
		
	
	
		Internat. J. Math. & Math. Sci
		
			8
			
			1986
		
	




* 
	
		?-closure and ?-subclosed graphs
		
			JEJoseph
		
	
	
		Math., Chronicle
		
			8
			
			1979
		
	




* 
	
		Extensions of topological spaces
		
			SFomin
		
	
	
		Ann. of Math
		
			44
			
			1943
		
	




* 
	
		The method of centred systems in the theory of topological spaces, Uspekhi Mat
		
			SIliadis
		
		
			SFomin
		
	
	
		Appl. Math
		
			21
			4
			
			1996. 1966. 2000
		
	
	Nauk.




* 
	
		On ?-continuity and strong ?-continuity
		
			MSaleh
		
	
	
		Applied Mathematics E-Notes
		
			
			2003
		
	




* 
	
		Some properties of ?-open sets
		
			MCaldas
		
		
			SJafari
		
		
			MMKovar
		
	
	
		Divulg. Mat
		
			12
			2
			
			2004
		
	




* 
	
		The ??-topology and faintly continuous functions
		
			PELong
		
		
			LLHerrington
		
	
	
		Kyungpook Math. J
		
			22
			
			1982
		
	




* 
	
		A note on some applications of ?-open sets
		
			MCaldas
		
	
	
		UMMS
		
			2
			
			2003
		
	




* 
	
		On ?-irresolute functions
		
			FHKhedr
		
		
			TNoiri
		
	
	
		Indian J. Math
		
			28
			
			1986
		
	




* 
	
		Introduction to ? -Type Transitive Maps on Topological spaces
		
			MuradMohammed Nokhas
		
	
	
		International Journal of Basic & Applied Sciences IJBAS-IJENS
		
			12
			06
			
			2012
		
	




* 
	
		
			MuradMohammed Nokhas
		
		Topologically -Transitive Maps and Minimal Systems Gen
				
	




* 
	
		
	
	
		Math. Notes
		2219-7184
		
			10
			2
			
			2012
		
	




* 
	
		
		
			IcsrsCopyright©
		
		
	




* 
	
		Outline of General Topology
		
			REngelking
		
		
			1968
			North Holland Publishing Company-Amsterdam
		
	




* 
	
		Conceptions of transitive maps in topological spaces
		
			MNokhas Murad Kaki
		
		
			SolafAliHussain
		
	
	
		International Journal of Electronics Communication and Computer Engineering
		(Online): 2249-071X
		
			5
			1
			
		
	
	Print
	ISSN




* 
	
		
			MuradMohammed Nokhas
		
	
	
		Relationship between New Types of Transitive Maps and Minimal Systems
				
			4
			
		
	
	Print
	ISSN




* 
	
		References Références Referencias
		
	




* 
	
		
			Mohammed Nokhas MuradKaki
		
		New types of chaotic maps on topological spaces, International Relationship between New Types of Transitive and Chaotic Maps ?